Dissipative phenomena manifest in multiple mechanical systems. In this dissertation, different geometric frameworks for modelling non-conservative dynamics are considered. The objective is to generalize several results from conservative systems to dissipative systems, specially those concerning the symmetries and integrability of these systems. More specifically, three classes of geometric frameworks modelling dissipative systems are considered: systems with external forces, contact systems and systems with impacts. The first two allow modelling a continuous dissipation of energy over time, while the latter also permits considering abrupt changes of energy in the instants of the impacts.
{"title":"The geometry of dissipation","authors":"Asier López-Gordón","doi":"arxiv-2409.11947","DOIUrl":"https://doi.org/arxiv-2409.11947","url":null,"abstract":"Dissipative phenomena manifest in multiple mechanical systems. In this\u0000dissertation, different geometric frameworks for modelling non-conservative\u0000dynamics are considered. The objective is to generalize several results from\u0000conservative systems to dissipative systems, specially those concerning the\u0000symmetries and integrability of these systems. More specifically, three classes\u0000of geometric frameworks modelling dissipative systems are considered: systems\u0000with external forces, contact systems and systems with impacts. The first two\u0000allow modelling a continuous dissipation of energy over time, while the latter\u0000also permits considering abrupt changes of energy in the instants of the\u0000impacts.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hokuto Konno, Jianfeng Lin, Anubhav Mukherjee, Juan Muñoz-Echániz
We study the monodromy diffeomorphism of Milnor fibrations of isolated complex surface singularities, by computing the family Seiberg--Witten invariant of Seifert-fibered Dehn twists using recent advances in monopole Floer homology. More precisely, we establish infinite order non-triviality results for boundary Dehn twists on indefinite symplectic fillings of links of minimally elliptic surface singularities. Using this, we exhibit a wide variety of new phenomena in dimension four: (1) smoothings of isolated complex surface singularities whose Milnor fibration has monodromy with infinite order as a diffeomorphism but with finite order as a homeomorphism, (2) robust Torelli symplectomorphisms that do not factor as products of Dehn--Seidel twists, (3) compactly supported exotic diffeomorphisms of exotic $mathbb{R}^4$'s and contractible manifolds.
{"title":"On four-dimensional Dehn twists and Milnor fibrations","authors":"Hokuto Konno, Jianfeng Lin, Anubhav Mukherjee, Juan Muñoz-Echániz","doi":"arxiv-2409.11961","DOIUrl":"https://doi.org/arxiv-2409.11961","url":null,"abstract":"We study the monodromy diffeomorphism of Milnor fibrations of isolated\u0000complex surface singularities, by computing the family Seiberg--Witten\u0000invariant of Seifert-fibered Dehn twists using recent advances in monopole\u0000Floer homology. More precisely, we establish infinite order non-triviality\u0000results for boundary Dehn twists on indefinite symplectic fillings of links of\u0000minimally elliptic surface singularities. Using this, we exhibit a wide variety\u0000of new phenomena in dimension four: (1) smoothings of isolated complex surface\u0000singularities whose Milnor fibration has monodromy with infinite order as a\u0000diffeomorphism but with finite order as a homeomorphism, (2) robust Torelli\u0000symplectomorphisms that do not factor as products of Dehn--Seidel twists, (3)\u0000compactly supported exotic diffeomorphisms of exotic $mathbb{R}^4$'s and\u0000contractible manifolds.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a new surgery type operation by switching between two exact fillings of Legendrians which we call a BSP surgery. In certain cases, this surgery can preserve monotonicity of Lagrangians. We prove a wall-crossing type formula for the change of the disk-potential under surgery with Bohr-Sommerfeld profiles. As an application, we show that Biran's circle-bundle lifts admit a Bohr-Sommerfeld type surgery. We use the wall-crossing theorem about disk-potentials to construct exotic monotone Lagrangian tori in $bP^n$.
{"title":"Bohr-Sommerfeld profile surgeries and Disk Potentials","authors":"Soham Chanda","doi":"arxiv-2409.11603","DOIUrl":"https://doi.org/arxiv-2409.11603","url":null,"abstract":"We construct a new surgery type operation by switching between two exact\u0000fillings of Legendrians which we call a BSP surgery. In certain cases, this\u0000surgery can preserve monotonicity of Lagrangians. We prove a wall-crossing type\u0000formula for the change of the disk-potential under surgery with Bohr-Sommerfeld\u0000profiles. As an application, we show that Biran's circle-bundle lifts admit a\u0000Bohr-Sommerfeld type surgery. We use the wall-crossing theorem about\u0000disk-potentials to construct exotic monotone Lagrangian tori in $bP^n$.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"98 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we develop a method to compute the Morse homology of a manifold when descending manifolds and ascending manifolds intersect cleanly, but not necessarily transversely. While obstruction bundle gluing defined by Hutchings and Taubes is a computable tool to handle non-transverse intersections, it has only been developed for specific cases. In contrast, most virtual techniques apply to general cases but lack computational efficiency. To address this, we construct minimal semi-global Kuranishi structures for the moduli spaces of Morse trajectories, which generalize obstruction bundle gluing while maintaining its computability feature. Through this construction, we obtain iterated gluing equals simultaneous gluing.
{"title":"Computable, obstructed Morse homology for clean intersections","authors":"Erkao Bao, Ke Zhu","doi":"arxiv-2409.11565","DOIUrl":"https://doi.org/arxiv-2409.11565","url":null,"abstract":"In this paper, we develop a method to compute the Morse homology of a\u0000manifold when descending manifolds and ascending manifolds intersect cleanly,\u0000but not necessarily transversely. While obstruction bundle gluing defined by Hutchings and Taubes is a\u0000computable tool to handle non-transverse intersections, it has only been\u0000developed for specific cases. In contrast, most virtual techniques apply to\u0000general cases but lack computational efficiency. To address this, we construct\u0000minimal semi-global Kuranishi structures for the moduli spaces of Morse\u0000trajectories, which generalize obstruction bundle gluing while maintaining its\u0000computability feature. Through this construction, we obtain iterated gluing\u0000equals simultaneous gluing.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1995, Cohen, Jones and Segal proposed a method of upgrading any given Floer homology to a stable homotopy-valued invariant. For a generic pseudo-gradient Morse-Bott flow on a closed smooth manifold $M$, we rigorously construct the conjectural stable normal framings, which are an essential ingredient in their construction, and give a rigorous proof that the resulting stable homotopy type recovers $Sigma^infty_+ M$. We further show that one can recover Thom spectra $M^E$ for all reduced $KO$-theory classes $E$ on $M$, by using slightly modified stable normal framings. Our paper also includes a construction of the smooth corner structure on compactified moduli spaces of broken flow lines with free endpoint, a formal construction of Piunikhin-Salamon-Schwarz type continuation maps, and a way to relax the stable normal framing condition to orientability in orthogonal spectra.
{"title":"Revisiting the Cohen-Jones-Segal construction in Morse-Bott theory","authors":"Ciprian Mircea Bonciocat","doi":"arxiv-2409.11278","DOIUrl":"https://doi.org/arxiv-2409.11278","url":null,"abstract":"In 1995, Cohen, Jones and Segal proposed a method of upgrading any given\u0000Floer homology to a stable homotopy-valued invariant. For a generic\u0000pseudo-gradient Morse-Bott flow on a closed smooth manifold $M$, we rigorously\u0000construct the conjectural stable normal framings, which are an essential\u0000ingredient in their construction, and give a rigorous proof that the resulting\u0000stable homotopy type recovers $Sigma^infty_+ M$. We further show that one can\u0000recover Thom spectra $M^E$ for all reduced $KO$-theory classes $E$ on $M$, by\u0000using slightly modified stable normal framings. Our paper also includes a\u0000construction of the smooth corner structure on compactified moduli spaces of\u0000broken flow lines with free endpoint, a formal construction of\u0000Piunikhin-Salamon-Schwarz type continuation maps, and a way to relax the stable\u0000normal framing condition to orientability in orthogonal spectra.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256828","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $X$ be a smooth complex manifold. Assume that $Ysubset X$ is a K"{a}hler submanifold such that $Xsetminus Y$ is biholomorphic to $mathbb{C}^n$. We prove that $(X, Y)$ is biholomorphic to the standard example $(mathbb{P}^n, mathbb{P}^{n-1})$. We then study certain K"{a}hler orbifold compactifications of $mathbb{C}^n$ and prove that on $mathbb{C}^3$ the flat metric is the only asymptotically conical Ricci-flat K"{a}hler metric whose metric cone at infinity has a smooth link. As a key technical ingredient, a new formula for minimal discrepancy of isolated Fano cone singularities in terms of generalized Conley-Zehnder indices in symplectic geometry is derived.
{"title":"Kähler compactification of $mathbb{C}^n$ and Reeb dynamics","authors":"Chi Li, Zhengyi Zhou","doi":"arxiv-2409.10275","DOIUrl":"https://doi.org/arxiv-2409.10275","url":null,"abstract":"Let $X$ be a smooth complex manifold. Assume that $Ysubset X$ is a\u0000K\"{a}hler submanifold such that $Xsetminus Y$ is biholomorphic to\u0000$mathbb{C}^n$. We prove that $(X, Y)$ is biholomorphic to the standard example\u0000$(mathbb{P}^n, mathbb{P}^{n-1})$. We then study certain K\"{a}hler orbifold\u0000compactifications of $mathbb{C}^n$ and prove that on $mathbb{C}^3$ the flat\u0000metric is the only asymptotically conical Ricci-flat K\"{a}hler metric whose\u0000metric cone at infinity has a smooth link. As a key technical ingredient, a new\u0000formula for minimal discrepancy of isolated Fano cone singularities in terms of\u0000generalized Conley-Zehnder indices in symplectic geometry is derived.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that homotopy cardinality -- a priori ill-defined for many dg-categories, including all periodic ones -- has a reasonable definition for even-dimensional Calabi--Yau (evenCY) categories and their relative generalizations (under appropriate finiteness conditions). As a first application we solve the problem of defining an intrinsic Hall algebra for degreewise finite pre-triangulated dg-categories in the case of oddCY categories. We compare this definition with To"en's derived Hall algebras (in case they are well-defined) and with other approaches based on extended Hall algebras and central reduction, including a construction of Hall algebras associated with Calabi--Yau triples of triangulated categories. For a category equivalent to the root category of a 1CY abelian category $mathcal A$, the algebra is shown to be isomorphic to the Drinfeld double of the twisted Ringel--Hall algebra of $mathcal A$, thus resolving in the Calabi--Yau case the long-standing problem of realizing the latter as a Hall algebra intrinsically defined for such a triangulated category. Our second application is the proof of a conjecture of Ng--Rutherford--Shende--Sivek, which provides an intrinsic formula for the ruling polynomial of a Legendrian knot $L$, and its generalization to Legendrian tangles, in terms of the augmentation category of $L$.
{"title":"Counting in Calabi--Yau categories, with applications to Hall algebras and knot polynomials","authors":"Mikhail Gorsky, Fabian Haiden","doi":"arxiv-2409.10154","DOIUrl":"https://doi.org/arxiv-2409.10154","url":null,"abstract":"We show that homotopy cardinality -- a priori ill-defined for many\u0000dg-categories, including all periodic ones -- has a reasonable definition for\u0000even-dimensional Calabi--Yau (evenCY) categories and their relative\u0000generalizations (under appropriate finiteness conditions). As a first application we solve the problem of defining an intrinsic Hall\u0000algebra for degreewise finite pre-triangulated dg-categories in the case of\u0000oddCY categories. We compare this definition with To\"en's derived Hall\u0000algebras (in case they are well-defined) and with other approaches based on\u0000extended Hall algebras and central reduction, including a construction of Hall\u0000algebras associated with Calabi--Yau triples of triangulated categories. For a\u0000category equivalent to the root category of a 1CY abelian category $mathcal\u0000A$, the algebra is shown to be isomorphic to the Drinfeld double of the twisted\u0000Ringel--Hall algebra of $mathcal A$, thus resolving in the Calabi--Yau case\u0000the long-standing problem of realizing the latter as a Hall algebra\u0000intrinsically defined for such a triangulated category. Our second application is the proof of a conjecture of\u0000Ng--Rutherford--Shende--Sivek, which provides an intrinsic formula for the\u0000ruling polynomial of a Legendrian knot $L$, and its generalization to\u0000Legendrian tangles, in terms of the augmentation category of $L$.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe some contact non-squeezing phenomena in lens spaces by defining and computing a contact capacity. This contact capacity comes from the spectral selectors constructed by Allais, Sandon and the author by the means of generating functions and Givental's non-linear Maslov index. We also discuss a potential generalization of these non-squeezing phenomena for orderable closed prequantizations.
{"title":"Contact non-squeezing in lens spaces","authors":"Pierre-Alexandre Arlove","doi":"arxiv-2409.10334","DOIUrl":"https://doi.org/arxiv-2409.10334","url":null,"abstract":"We describe some contact non-squeezing phenomena in lens spaces by defining\u0000and computing a contact capacity. This contact capacity comes from the spectral\u0000selectors constructed by Allais, Sandon and the author by the means of\u0000generating functions and Givental's non-linear Maslov index. We also discuss a\u0000potential generalization of these non-squeezing phenomena for orderable closed\u0000prequantizations.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a symplectic 4-manifold $M$ equipped with a singular Lagrangian fibration with a section, the natural fiberwise addition given by the local Hamiltonian flow is well-defined on the regular points. We prove, in the case that the singularities are of focus-focus type, that the closure of the corresponding addition graph is the image of a Lagrangian immersion in $(M times M)^- times M$, and we study its geometry. Our main motivation for this result is the construction of a symmetric monoidal structure on the Fukaya category of such a manifold.
{"title":"The focus-focus addition graph is immersed","authors":"Mohammed Abouzaid, Nathaniel Bottman, Yunpeng Niu","doi":"arxiv-2409.10377","DOIUrl":"https://doi.org/arxiv-2409.10377","url":null,"abstract":"For a symplectic 4-manifold $M$ equipped with a singular Lagrangian fibration\u0000with a section, the natural fiberwise addition given by the local Hamiltonian\u0000flow is well-defined on the regular points. We prove, in the case that the\u0000singularities are of focus-focus type, that the closure of the corresponding\u0000addition graph is the image of a Lagrangian immersion in $(M times M)^- times\u0000M$, and we study its geometry. Our main motivation for this result is the\u0000construction of a symmetric monoidal structure on the Fukaya category of such a\u0000manifold.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper explores the structure of bi-Lagrangian Grassmanians for pencils of $2$-forms on real or complex vector spaces. We reduce the analysis to the pencils whose Jordan-Kronecker Canonical Form consists of Jordan blocks with the same eigenvalue. We demonstrate that this is equivalent to studying Lagrangian subspaces invariant under a nilpotent self-adjoint operator. We calculate the dimension of bi-Lagrangian Grassmanians and describe their open orbit under the automorphism group. We completely describe the automorphism orbits in the following three cases: for one Jordan block, for sums of equal Jordan blocks and for a sum of two distinct Jordan blocks.
{"title":"Geometry of bi-Lagrangian Grassmannian","authors":"I. K. Kozlov","doi":"arxiv-2409.09855","DOIUrl":"https://doi.org/arxiv-2409.09855","url":null,"abstract":"This paper explores the structure of bi-Lagrangian Grassmanians for pencils\u0000of $2$-forms on real or complex vector spaces. We reduce the analysis to the\u0000pencils whose Jordan-Kronecker Canonical Form consists of Jordan blocks with\u0000the same eigenvalue. We demonstrate that this is equivalent to studying\u0000Lagrangian subspaces invariant under a nilpotent self-adjoint operator. We\u0000calculate the dimension of bi-Lagrangian Grassmanians and describe their open\u0000orbit under the automorphism group. We completely describe the automorphism\u0000orbits in the following three cases: for one Jordan block, for sums of equal\u0000Jordan blocks and for a sum of two distinct Jordan blocks.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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